Question

$\Large{\bf{\green{\mathfrak{\dag{\underline{\underline{Given:-}}}}}}}$

| x | + y = 4

x + 3 | y | = 6

Find out the values of x and y.

$\orange{\textsf{Answers:-}}$ (3 , 1) ; $\sf \bigg( \dfrac{- \: 3}{2} \: , \: \dfrac{5}{2} \bigg)$ ; (- 9 , - 5) ; $\sf \bigg( \dfrac{9}{2} \: , \: \dfrac{- \: 1}{2} \bigg)$

I want complete explanation with correct steps.

@Takenname, mods, stars and users help me.

$\purple{\textsf{Topic:-}}$ Moduless function

$\red{\textsf{Don't Spam}}$​

Answer 1

EXPLANATION.

⇒ |x| + y = 4. - - - - - (1).

⇒ x + 3|y| = 6. - - - - - (2).

As we know that,

Case = 1.

⇒ |x| ≥ 0.

⇒ x + y = 4. - - - - - (1).

⇒ x + 3y = 6. - - - - - (2).

Subtract both equation (1) & (2), we get.

⇒ - 2y = - 2.

⇒ y = 1.

Put the value of y = 1 in equation (1), we get.

⇒ x + 1 = 4.

⇒ x = 4 - 1.

⇒ x = 3.

Values of x = 3 & y = 1.

Case = 2.

⇒ |x| ≤ 0.

⇒ - x + y = 4. - - - - - (1).

⇒ x - 3y = 6. - - - - - (2).

From equation (1) & (2), we get.

⇒ - 2y = 10.

⇒ y = - 5.

Put the value of y = - 5in equation (2), we get.

⇒ x - 3(-5) = 6.

⇒ x + 15 = 6.

⇒ x = 6 - 15.

⇒ x = - 9.

Values of x = - 9 & y = - 5.

Case = 3.

⇒ |x| = - a ≤ x ≤ a.

Apply modulus function on equation (1), we get.

x ≤ a.

⇒ - x + y = 4. - - - - - (1).

⇒ x + 3y = 6. - - - - - (2).

From equation (1) & (2), we get.

⇒ 4y = 10.

⇒ y = 10/4 = 5/2.

Put the value of y = 5/2 in equation (2), we get.

⇒ x + 3(5/2) = 6.

⇒ x + 15/2 = 6.

⇒ x = 6 - 15/2.

⇒ x = 12 - 15/2.

⇒ x = - 3/2.

Values of x = -3/2 & y = 5/2.

Case = 4.

Apply modulus function on equation (2), we get.

x ≤ a.

⇒ x + y = 4. - - - - - (1).

⇒ x - 3y = 6. - - - - - (2).

From equation (1) & (2), we get.

Subtract both the equation, we get.

⇒ 4y = - 2.

⇒ y = -1/2.

Put the value of y = -1/2 in equation (1), we get.

⇒ x - 1/2 = 4.

⇒ x = 4 + 1/2.

⇒ x = 8 + 1/2.

⇒ x = 9/2.

Values of x = 9/2 & y = -1/2.

                                                                                                                   

MORE INFORMATION.

Properties of modulus function.

(1) = |x| ≤ a ⇒ - a ≤ x ≤ a.

(2) = |x| ≥ a ⇒ x ≤ - a Or x ≥ a.

(3) = |x + y| = |x| + |y| ⇒ x, y ≥ 0 Or x ≤ 0, y ≤ 0.

(4) = |x - y| = |x| - |y| ⇒ x ≥ 0 And |x| ≥ |y| Or x ≤ 0 And y ≤ 0 And |x| ≥ |y|.

(5) = |x ± y| ≤ |x| + |y|.

(6) = |x + y| ≥ |x| - |y|.

(7) = |x - y| ≥ |x| - |y|.

Answer 2

Answer:

$\Large{\bf{\green{\mathfrak{\dag{\underline{\underline{Given:-}}}}}}}$

| x | + y = 4

x + 3 | y | = 6

Find out the values of x and y.

$\orange{\textsf{Answers:-}}$ (3 , 1) ; $\sf \bigg( \dfrac{- \: 3}{2} \: , \: \dfrac{5}{2} \bigg)$ ; (- 9 , - 5) ; $\sf \bigg( \dfrac{9}{2} \: , \: \dfrac{- \: 1}{2} \bigg)$

I want complete explanation with correct steps.

@Takenname, mods, stars and users help me.

$\purple{\textsf{Topic:-}}$ Moduless function

$\red{\textsf{Don't Spam}}$